📝

Simplifying Boolean Expressions with K-maps

May 7, 2024

View transcript

Lecture Summary

In this lecture, we explored how to convert the truth table data into a logical function using Karnaugh Maps (K-maps). We covered the basics of creating and populating a K-map for three and four variable functions, and demonstrated how to simplify Boolean expressions using grouping in the K-map. We then further illustrated how to convert these simplified expressions into circuit diagrams through AND, OR, and NOT gates.

Detailed Notes

Understanding Karnaugh Maps (K-maps)

  1. Basics of Karnaugh Maps (K-maps)

    • Karnaugh maps simplify Boolean algebra expressions.
    • They help visually identify commonalities in truth tables.
    • K-maps consist of a grid where each cell represents a variable combination and its resultant output.

  2. Creating a 3-Variable K-map

    • A 3-variable K-map has 8 cells (2^3 for three variables: A, B, C).
    • Arranged in two rows and four columns.
    • Cells are populated based on the outputs for combinations of A, B, and C from the truth table.

  3. Populating a 3-Variable K-map

    • For variable ABC: Assign rows to C (C=0 top, C=1 bottom), columns split into groups by AB (00, 01, 11, 10).
    • Enter output values from the truth table into corresponding cells.

  4. Simplifying Boolean Functions using K-map Groups

    • Groups of 1's in the K-map are used to simplify the expression.
    • Circle groups of 1's (power of 2 size: 1, 2, 4...).
    • Create simplified product terms where variables remain constant within each group and remove (ignore) variables that change within the group.

  5. Boolean Functions from 3-Variable K-map Example

    • From groups, deduce minimal expressions: (B AND NOT C) OR (A AND NOT B).

From K-map to Circuit Diagram

  1. Interpreting Functions from Groups

    • Extract functions: E.g., Function = B(NOT C) + A(NOT B).

  2. Drawing Circuit Diagrams

    • Use AND gates for each product term and tie them with OR gate based on the derived function.
    • For each product term: Connect corresponding inputs to AND gates, each output of these AND gates go to an OR gate to produce the function output.

4-Variable K-map Example

  1. Setting Up

    • A 4-variable K-map has 16 cells (2^4 for variables A, B, C, D).
    • Organized into four rows and four columns for combinations of CD and AB respectively.

  2. Simplifying Functions

    • Similar process: Identify and circle groups of 1's. Simplify expressions by observing which variables change across the group.
    • Create outputs in terms of AND (product terms), and tie them together in OR (sum terms).

  3. Circuit Implementation

    • For each product term, assign an AND gate.
    • Combine outputs with an OR gate for final function output.
    • Use NOT gates for complemented variables.

Practical Examples and Exercises

  • Walked through filling a 4-variable K-map with another truth table.
  • Discussed various configurations for simplifying using different variable groupings.
  • Translated resulting Boolean expressions back into circuit diagrams."

Summary

Through this lecture, we learned the practical applications of Karnaugh Maps in simplifying Boolean expressions and converting them into circuit diagrams, crucial for efficient digital circuit design. This methodology provides a visual and systematic approach to handling logic functions derived from truth tables.

Full transcript